Flint2-0.1.0.5: Haskell bindings for the flint library for number theory

Description

A Qadic represents an element of $$\mathbb{Q}_q \cong \mathbb{Q}_p[X] / (f(X))$$. This module implements operations on q-adic numbers.

Example

Calculate a root of the polynomial $$x^{10}+10x^9+9x^8+8x^7+8x^6+2x^4+9x^3+x^2+3x+1$$ over $$K=\mathbb{Q}_{{11}^4} \cong \mathbb{Q}_{11}[X] /(X^4+8X^2+10X+2)$$ to standard padic precision using Newton iteration. The iteration starts with $$x=8a^3+4a^2+3$$ where $$a$$ is a generator of $$K$$. The value of $$x$$ is initialized using a FmpzPoly.

import Data.Number.Flint

main = do
let c = [1,10,9,8,8,0,2,9,1,3,1]
withNewQadicCtx 11 4 0 128 "a" padic_series $\ctx -> do CQadicCtx pctx _ _ _ _ <- peek ctx withNewQadic$ \x -> do
withFmpzPoly (fromList [3,0,4,8]) $\poly -> do padic_poly_set_fmpz_poly x poly pctx newton x c ctx putStr "x = " qadic_print_pretty x ctx putStr "\n" y <- horner x c ctx withQadic y$ \y -> do
putStr "y = "
putStr "\n"

newton x c ctx = do
withNewQadic $\y -> withNewQadic$ \y' -> do
withNewQadic $\tmp -> forM_ (tail c)$ \c -> do
when (is_zero /= 1) $newton x c ctx return () horner x c ctx = do y <- newQadic withQadic y$ \y -> do
withNewQadic $\tmp -> forM_ (tail c)$ \c -> do
return y



Running main yields:

>>> main
x = (8*a^3+4*a^2+3) + (8*a^2+2*a+5)*11 + (8*a^3+a^2+6)*11^2 + (7*a^3+6*a^2+2*a+6)*11^3 + (10*a^3+6*a^2+9*a+3)*11^4 + (6*a^3+6*a^2+3*a+7)*11^5 + (7*a^3+5*a^2+9*a+9)*11^6 + (2*a^2+4*a+3)*11^7 + (a^3+3*a^2+3*a+8)*11^8 + (2*a^3+2*a^2+8*a+2)*11^9 + (5*a^3+9*a^2)*11^10 + (2*a^3+3*a^2+2*a+7)*11^11 + (a^3+4*a^2+7*a+3)*11^12 + (10*a^3+9*a^2+10*a+6)*11^13 + (7*a^3+a^2+9*a+3)*11^14 + (10*a^3+10*a^2+6*a+4)*11^15 + (3*a^3+a^2+2*a+1)*11^16 + (4*a^3+6*a^2+8*a)*11^17 + (2*a^3+9*a^2+9*a+10)*11^18 + (4*a^3+4*a^2+5*a+4)*11^19

Synopsis

Data structures We represent an element of the extension $$\mathbb{Q}_q \cong \mathbb{Q}_p[X]\ /\ f(X)$$ as a polynomial in $$\mathbb{Q}_p[X]$$ of degree less than $$\deg(f)$$. As such, qadic_struct and qadic_t are typedef'ed as padic_poly_struct and padic_poly_t.

Constructors

Apply f to new q-adic

Apply f to new q-adic

Context We represent an unramified extension of $$\mathbb{Q}_p$$ via $$\mathbb{Q}_q \cong \mathbb{Q}_p[X]\ /\ f(X)$$, where $$f \in \mathbb{Q}_p[X]$$ is a monic, irreducible polynomial which we assume to actually be in $$\mathbb{Z}[X]$$. The first field in the context structure is a $$p$$-adic context struct pctx, which contains data about the prime $$p$$, precomputed powers, the printing mode etc. The polynomial $$f$$ is represented as a sparse polynomial using two arrays $$j$$ and $$a$$ of length len, where $$f(X) = \sum_{i} a_{i} X^{j_{i}}$$. We also assume that the array $$j$$ is sorted in ascending order. We choose this data structure to improve reduction modulo $$f(X)$$ in $$\mathbb{Q}_p[X]$$, assuming a sparse polynomial $$f(X)$$ is chosen. The field var contains the name of a generator of the extension, which is used when printing the elements.

Constructors

Constructors

Instances

Instances details
 Source # Instance detailsDefined in Data.Number.Flint.Qadic.FFI MethodspokeElemOff :: Ptr CQadicCtx -> Int -> CQadicCtx -> IO () #peekByteOff :: Ptr b -> Int -> IO CQadicCtx #pokeByteOff :: Ptr b -> Int -> CQadicCtx -> IO () #poke :: Ptr CQadicCtx -> CQadicCtx -> IO () #

Create q-adic context with prime $$p$$, extension $$d$$, precomputed powers $$p^{min}$$ to $$p^{max}$$ and PadicPrintMode mode. Initialized with qadic_ctx_init.

Create q-adic context with prime $$p$$, extension $$d$$, precomputed powers $$p^{min}$$ to $$p^{max}$$ and PadicPrintMode mode. Initialized with qadic_ctx_init_conway.

qadic_ctx_init ctx p d min max var mode

Initialises the context ctx with prime $$p$$, extension degree $$d$$, variable name var and printing mode mode. The defining polynomial is chosen as a Conway polynomial if possible and otherwise as a random sparse polynomial.

Stores powers of $$p$$ with exponents between min (inclusive) and max exclusive. Assumes that min is at most max.

Assumes that $$p$$ is a prime.

Assumes that the string var is a null-terminated string of length at least one.

Assumes that the printing mode is one of PADIC_TERSE, PADIC_SERIES, or PADIC_VAL_UNIT.

This function also carries out some relevant precomputation for arithmetic in $$\mathbb{Q}_p / (p^N)$$ such as powers of $$p$$ close to $$p^N$$.

qadic_ctx_init_conway ctx p d min max var mode

Initialises the context ctx with prime $$p$$, extension degree $$d$$, variable name var and printing mode mode. The defining polynomial is chosen as a Conway polynomial, hence has restrictions on the prime and the degree.

Stores powers of $$p$$ with exponents between min (inclusive) and max exclusive. Assumes that min is at most max.

Assumes that $$p$$ is a prime.

Assumes that the string var is a null-terminated string of length at least one.

Assumes that the printing mode is one of PADIC_TERSE, PADIC_SERIES, or PADIC_VAL_UNIT.

This function also carries out some relevant precomputation for arithmetic in $$\mathbb{Q}_p / (p^N)$$ such as powers of $$p$$ close to $$p^N$$.

Clears all memory that has been allocated as part of the context.

Returns the extension degree.

Prints the data from the given context.

Memory management

Initialises the element rop, setting its value to $$0$$.

Initialises the element rop with the given output precision, setting the value to $$0$$.

Clears the element rop.

_fmpz_poly_reduce R lenR a j len

Reduces a polynomial (R, lenR) modulo a sparse monic polynomial $$f(X) = \sum_{i} a_{i} X^{j_{i}}$$ of degree at least $$2$$.

Assumes that the array $$j$$ of positive length len is sorted in ascending order.

Allows zero-padding in (R, lenR).

_fmpz_mod_poly_reduce R lenR a j len p

Reduces a polynomial (R, lenR) modulo a sparse monic polynomial $$f(X) = \sum_{i} a_{i} X^{j_{i}}$$ of degree at least $$2$$ in $$\mathbb{Z}/(p)$$, where $$p$$ is typically a prime power.

Assumes that the array $$j$$ of positive length len is sorted in ascending order.

Allows zero-padding in (R, lenR).

Reduces rop modulo $$f(X)$$ and $$p^N$$.

Properties

Returns the valuation of op.

Returns the precision of op.

Randomisation

Generates a random element of $$\mathbb{Q}_q$$.

Generates a random non-zero element of $$\mathbb{Q}_q$$.

Generates a random element of $$\mathbb{Q}_q$$ with prescribed valuation val.

Note that if $$v \geq N$$ then the element is necessarily zero.

Generates a random element of $$\mathbb{Q}_q$$ with non-negative valuation.

Assignments and conversions

Sets rop to op.

Sets rop to zero.

Sets rop to one, reduced in the given context.

Note that if the precision $$N$$ is non-positive then rop is actually set to zero.

Sets rop to the generator $$X$$ for the extension when $$N > 0$$, and zero otherwise. If the extension degree is one, raises an abort signal.

Sets rop to the integer op, reduced in the context.

If the element op lies in $$\mathbb{Q}_p$$, sets rop to its value and returns $$1$$; otherwise, returns $$0$$.

Comparison

Returns whether op is equal to zero.

Returns whether op is equal to one in the given context.

Returns whether op1 and op2 are equal.

Basic arithmetic

Sets rop to the sum of op1 and op2.

Assumes that both op1 and op2 are reduced in the given context and ensures that rop is, too.

Sets rop to the difference of op1 and op2.

Assumes that both op1 and op2 are reduced in the given context and ensures that rop is, too.

Sets rop to the negative of op.

Assumes that op is reduced in the given context and ensures that rop is, too.

Sets rop to the product of op1 and op2, reducing the output in the given context.

_qadic_inv :: Ptr CFmpz -> Ptr CFmpz -> CLong -> Ptr CFmpz -> Ptr CLong -> CLong -> Ptr CFmpz -> CLong -> IO () Source #

_qadic_inv rop op len a j lena p N

Sets (rop, d) to the inverse of (op, len) modulo $$f(X)$$ given by (a,j,lena) and $$p^N$$.

Assumes that (op,len) has valuation $$0$$, that is, that it represents a $$p$$-adic unit.

Assumes that len is at most $$d$$.

Does not support aliasing.

Sets rop to the inverse of op, reduced in the given context.

_qadic_pow :: Ptr CFmpz -> Ptr CFmpz -> CLong -> Ptr CFmpz -> Ptr CFmpz -> Ptr CLong -> CLong -> Ptr CFmpz -> IO () Source #

_qadic_pow rop op len e a j lena p

Sets (rop, 2*d-1) to (op,len) raised to the power $$e$$, reduced modulo $$f(X)$$ given by (a, j, lena) and $$p$$, which is expected to be a prime power.

Assumes that $$e \geq 0$$ and that len is positive and at most $$d$$.

Although we require that rop provides space for $$2d - 1$$ coefficients, the output will be reduces modulo $$f(X)$$, which is a polynomial of degree $$d$$.

Does not support aliasing.

Sets rop the op raised to the power $$e$$.

Currently assumes that $$e \geq 0$$.

Note that for any input op, rop is set to one in the given context whenever $$e = 0$$.

Square root

Return 1 if the input is a square (to input precision). If so, set rop to a square root (truncated to output precision).

Special functions

_qadic_exp_rectangular rop op v len a j lena p N pN

Sets (rop, 2*d - 1) to the exponential of (op, v, len) reduced modulo $$p^N$$, assuming that the series converges.

Assumes that (op, v, len) is non-zero.

Does not support aliasing.

Returns whether the exponential series converges at op and sets rop to its value reduced modulo in the given context.

_qadic_exp_balanced :: Ptr CFmpz -> Ptr CFmpz -> CLong -> CLong -> Ptr CFmpz -> Ptr CLong -> CLong -> Ptr CFmpz -> CLong -> Ptr CFmpz -> IO () Source #

_qadic_exp_balanced rop x v len a j lena p N pN

Sets (rop, d) to the exponential of (op, v, len) reduced modulo $$p^N$$, assuming that the series converges.

Assumes that len is in $$[1,d)$$ but supports zero padding, including the special case when (op, len) is zero.

Supports aliasing between rop and op.

Returns whether the exponential series converges at op and sets rop to its value reduced modulo in the given context.

_qadic_exp :: Ptr CFmpz -> Ptr CFmpz -> CLong -> CLong -> Ptr CFmpz -> Ptr CLong -> CLong -> Ptr CFmpz -> CLong -> IO () Source #

_qadic_exp rop op v len a j lena p N

Sets (rop, 2*d - 1) to the exponential of (op, v, len) reduced modulo $$p^N$$, assuming that the series converges.

Assumes that (op, v, len) is non-zero.

Does not support aliasing.

Returns whether the exponential series converges at op and sets rop to its value reduced modulo in the given context.

The exponential series converges if the valuation of op is at least $$2$$ or $$1$$ when $$p$$ is even or odd, respectively.

_qadic_log_rectangular z y v len a j lena p N pN

Computes

$$ $z = - \sum_{i = 1}^{\infty} \frac{y^i}{i} \pmod{p^N}.$

Note that this can be used to compute the $$p$$-adic logarithm via the equation

$$ \begin{aligned} \log(x) & = \sum_{i=1}^{\infty} (-1)^{i-1} \frac{(x-1)^i}{i} \\ & = - \sum_{i=1}^{\infty} \frac{(1-x)^i}{i}. \end{aligned}

Assumes that $$y = 1 - x$$ is non-zero and that $$v = \operatorname{ord}_p(y)$$ is at least $$1$$ when $$p$$ is odd and at least $$2$$ when $$p = 2$$ so that the series converges.

Assumes that $$y$$ is reduced modulo $$p^N$$.

Assumes that $$v < N$$, and in particular $$N \geq 2$$.

Supports aliasing between $$y$$ and $$z$$.

Returns whether the $$p$$-adic logarithm function converges at op, and if so sets rop to its value.

_qadic_log_balanced z y len a j lena p N pN

Computes $$(z, d)$$ as

$$ $z = - \sum_{i = 1}^{\infty} \frac{y^i}{i} \pmod{p^N}.$

Assumes that $$v = \operatorname{ord}_p(y)$$ is at least $$1$$ when $$p$$ is odd and at least $$2$$ when $$p = 2$$ so that the series converges.

Supports aliasing between $$z$$ and $$y$$.

Returns whether the $$p$$-adic logarithm function converges at op, and if so sets rop to its value.

_qadic_log :: Ptr CFmpz -> Ptr CFmpz -> CLong -> CLong -> Ptr CFmpz -> Ptr CLong -> CLong -> Ptr CFmpz -> CLong -> Ptr CFmpz -> IO () Source #

_qadic_log z y v len a j lena p N pN

Computes $$(z, d)$$ as

$$ $z = - \sum_{i = 1}^{\infty} \frac{y^i}{i} \pmod{p^N}.$

Note that this can be used to compute the $$p$$-adic logarithm via the equation

$$ \begin{aligned} \log(x) & = \sum_{i=1}^{\infty} (-1)^{i-1} \frac{(x-1)^i}{i} \\ & = - \sum_{i=1}^{\infty} \frac{(1-x)^i}{i}. \end{aligned}

Assumes that $$y = 1 - x$$ is non-zero and that $$v = \operatorname{ord}_p(y)$$ is at least $$1$$ when $$p$$ is odd and at least $$2$$ when $$p = 2$$ so that the series converges.

Assumes that $$(y, d)$$ is reduced modulo $$p^N$$.

Assumes that $$v < N$$, and hence in particular $$N \geq 2$$.

Supports aliasing between $$z$$ and $$y$$.

Returns whether the $$p$$-adic logarithm function converges at op, and if so sets rop to its value.

The $$p$$-adic logarithm function is defined by the usual series

$$ $\log_p(x) = \sum_{i=1}^{\infty} (-1)^{i-1} \frac{(x-1)^i}{i}$

but this only converges when $$\operatorname{ord}_p(x)$$ is at least $$2$$ or $$1$$ when $$p = 2$$ or $$p > 2$$, respectively.

_qadic_frobenius_a rop e a j lena p N

Computes $$\sigma^e(X) \bmod{p^N}$$ where $$X$$ is such that $$\mathbb{Q}_q \cong \mathbb{Q}_p[X]/(f(X))$$.

Assumes that the precision $$N$$ is at least $$2$$ and that the extension is non-trivial, i.e.(d geq 2).

Assumes that $$0 < e < d$$.

Sets (rop, 2*d-1), although the actual length of the output will be at most $$d$$.

_qadic_frobenius :: Ptr CFmpz -> Ptr CFmpz -> CLong -> CLong -> Ptr CFmpz -> Ptr CLong -> CLong -> Ptr CFmpz -> CLong -> IO () Source #

_qadic_frobenius rop op len e a j lena p N

Sets (rop, 2*d-1) to $$\Sigma$$ evaluated at (op, len).

Assumes that len is positive but at most $$d$$.

Assumes that $$0 < e < d$$.

Does not support aliasing.

Evaluates the homomorphism $$\Sigma^e$$ at op.

Recall that $$\mathbb{Q}_q / \mathbb{Q}_p$$ is Galois with Galois group $$\langle \Sigma \rangle \cong \langle \sigma \rangle$$, which is also isomorphic to $$\mathbb{Z}/d\mathbb{Z}$$, where $$\sigma \in \operatorname{Gal}(\mathbb{F}_q/\mathbb{F}_p)$$ is the Frobenius element $$\sigma \colon x \mapsto x^p$$ and $$\Sigma$$ is its lift to $$\operatorname{Gal}(\mathbb{Q}_q/\mathbb{Q}_p)$$.

This functionality is implemented as GaloisImage() in Magma.

_qadic_teichmuller rop op len a j lena p N

Sets (rop, d) to the Teichm"uller lift of (op, len) modulo $$p^N$$.

Does not support aliasing.

Sets rop to the Teichm"uller lift of op to the precision given in the context.

For a unit op, this is the unique ((q-1)th root of unity which is congruent to )op modulo :math:p.

Sets rop to zero if op is zero in the given context.

Raises an exception if the valuation of op is negative.

_qadic_trace :: Ptr CFmpz -> Ptr CFmpz -> CLong -> Ptr CFmpz -> Ptr CLong -> CLong -> Ptr CFmpz -> IO () Source #

_qadic_trace rop op len a j lena pN

Sets rop to the trace of op.

For an element $$a \in \mathbb{Q}_q$$, multiplication by $$a$$ defines a $$\mathbb{Q}_p$$-linear map on $$\mathbb{Q}_q$$. We define the trace of $$a$$ as the trace of this map. Equivalently, if $$\Sigma$$ generates $$\operatorname{Gal}(\mathbb{Q}_q / \mathbb{Q}_p)$$ then the trace of $$a$$ is equal to $$\sum_{i=0}^{d-1} \Sigma^i (a)$$.

_qadic_norm :: Ptr CFmpz -> Ptr CFmpz -> CLong -> Ptr CFmpz -> Ptr CLong -> CLong -> Ptr CFmpz -> CLong -> IO () Source #

_qadic_norm rop op len a j lena p N

Sets rop to the norm of the element (op,len) in $$\mathbb{Z}_q$$ to precision $$N$$, where len is at least one.

The result will be reduced modulo $$p^N$$.

Note that whenever (op,len) is a unit, so is its norm. Thus, the output rop of this function will typically not have to be canonicalised or reduced by the caller.

Computes the norm of op to the given precision.

Algorithm selection is automatic depending on the input.

Whenever op has valuation greater than $$(p-1)^{-1}$$, this routine computes its norm rop via

$$ $\operatorname{Norm} (x) = \exp \Bigl( \bigl( \operatorname{Trace} \log (x) \bigr) \Bigr).$

In the special case that op lies in $$\mathbb{Q}_p$$, returns its norm as $$\operatorname{Norm}(x) = x^d$$, where $$d$$ is the extension degree.

Otherwise, raises an abort signal.

The complexity of this implementation is quasi-linear in $$d$$ and $$N$$, and polynomial in $$\log p$$.

Sets rop to the norm of op, using the formula

$$ $\operatorname{Norm}(x) = \ell(f)^{-\deg(a)} \operatorname{Res}(f(X), a(X)),$

where $$\mathbb{Q}_q \cong \mathbb{Q}_p[X] / (f(X))$$, $$\ell(f)$$ is the leading coefficient of $$f(X)$$, and $$a(X) \in \mathbb{Q}_p[X]$$ denotes the same polynomial as $$x$$.

The complexity of the current implementation is given by $$\mathcal{O}(d^4 M(N \log p))$$, where $$M(n)$$ denotes the complexity of multiplying to $$n$$-bit integers.

Output

Prints a pretty representation of op to file.
In the current implementation, always returns $$1$$. The return code is part of the function's signature to allow for a later implementation to return the number of characters printed or a non-positive error code.
Prints a pretty representation of op to stdout.
In the current implementation, always returns $$1$$. The return code is part of the function's signature to allow for a later implementation to return the number of characters printed or a non-positive error code.